3.3.2 Apply the sum and difference rules to combine derivatives. 3.3.3 Use the product rule for finding the derivative of a product of functions. 3.3.4 Use the quotient rule for finding the derivative of a quotient of functions. 3.3.5 Extend the power rule to functions with negative exponents. Given two differentiable functions, the quotient rule can be used to determine the derivative of the ratio of the two functions,. This can also be written as. The quotient rule is as follows: Example.

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Example 4---Derivatives of Radicals

Suppose $$displaystyle f(x) = sqrt[4] x + frac 6 {sqrt x}$$. Find $$f'(x)$$.

Step 1

• Quotient Rule Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.
• The quotient rule is used to determine the derivative of one function divided by another.
• For notes and practice problems, visit the Calculus course on (Version #1) is created for a 45-minute class period and fo.

Rewrite the function so each term is a power function (i.e., has the form $$ax^n$$).

$$ begin{align*} f(x) & = sqrt[4] x + frac 6 {sqrt x}[6pt] & = x^{1/4} + frac 6 {x^{1/2}}[6pt] & = x^{1/4} + 6x^{-1/2} end{align*} $$

Step 2

Use the power rule for derivatives to differentiate each term.

$$ begin{align*} f(x) & = x^{blue{1/4}} + 6x^{red{-1/2}}[6pt] & = blue{frac 1 4} x^{blue{frac 1 4} - 1} + 6red{left(-frac 1 2right)}x^{red{-frac 1 2} -1}[6pt] & = frac 1 4 x^{frac 1 4 - frac 4 4} - 3x^{-frac 1 2 - frac 2 2}[6pt] & = frac 1 4 x^{-3/4} - 3x^{-3/2} end{align*} $$

Step 3 (Optional)

Since the original function was written in terms of radicals, we rewrite the derivative in terms of radicals as well so they match aesthetically.

$$ begin{align*} f'(x) & = frac 1 4 x^{-3/4} - 3x^{-3/2}[6pt] & = frac 1 4cdot frac 1 {x^{3/4}} - 3cdot frac 1 {x^{3/2}}[6pt] & = frac 1 4 cdot frac 1 {sqrt[4]{x^3}} - frac 3 {sqrt{x^3}}[6pt] & frac 1 {4sqrt[4]{x^3}} - frac 3 {xsqrt x} end{align*} $$

If we rationalize the denominators as well we end up with

$$f'(x) = frac{sqrt[4] x}{4x} - frac{3sqrt x}{x^2}$$

Answer

$$displaystyle f'(x) = frac 1 {4sqrt[4]{x^3}} - frac 3 {xsqrt x} = frac{sqrt[4] x}{4x} - frac{3sqrt x}{x^2}$$ when $$displaystyle f(x) = sqrt[4] x + frac 6 {sqrt x}$$.

1,001 Calculus Practice Problems

Part I

The Questions

Chapter 5

This chapter focuses on some of the major techniques needed to find the derivative: the product rule, the quotient rule, and the chain rule. By using these rules along with the power rule and some basic formulas (see Chapter 4), you can find the derivatives of most of the single-variable functions you encounter in calculus. However, after using the derivative rules, you often need many algebra steps to simplify the function so that it's in a nice final form, especially on problems involving the product rule or quotient rule.

The Problems You'll Work On

Here you practice using most of the techniques needed to find derivatives (besides the power rule):

· The product rule

· The quotient rule

· The chain rule

· Derivatives involving trigonometric functions

What to Watch Out For

Many of these problems require one calculus step and then many steps of algebraic simplification to get to the final answer. Remember the following tips as you work through the problems:

· Considering simplifying a function before taking the derivative. Simplifying before taking the derivative is almost always easier than finding the derivative and then simplifying.

· Some problems have functions without specified formulas in the questions; don't be thrown off! Simply proceed as you normally would on a similar example.

· Many people make the mistake of using the product rule when they should be using the chain rule. Stop and examine the function before jumping in and taking the derivative. Make sure you recognize whether the question involves a product or a composition (in which case you must use the chain rule).

· Rewriting the function by adding parentheses or brackets may be helpful, especially on problems that involve using the chain rule multiple times.

Using the Product Rule to Find Derivatives

312–331 Use the product rule to find the derivative of the given function.

312.f (x) = (2x³ + 1)(x⁵ – x)

313.f (x) = x² sin x

314.f (x) = sec x tan x

315.

316.f (x) = 4x csc x

317. Find (fg)'(4) if f (4) = 3, f '(4) = 2, g(4) = –6, and g '(4) = 8.

318.

319.f (x) = (sec x)(x + tan x)

320.

321.f (x) = 4x³ sec x

322.

323. Assuming that g is a differentiable function, find an expression for the derivative of f (x) = x²g(x).

324. Assuming that g is a differentiable function, find an expression for the derivative of .

325. Find (fg)'(3) if f (3) = –2, f '(3) = 4, g(3) = –8, and g '(3) = 7.

326.f (x) = x² cos x sin x

327. Assuming that g is a differentiable function, find an expression for the derivative of .

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328.

329.

330. Assuming that g is a differentiable function, find an expression for the derivative of .

331. Assuming that g and h are differentiable functions, find an expression for the derivative of .

Using the Quotient Rule to Find Derivatives

332–351 Use the quotient rule to find the derivative.

332.

333.

334.

335.

336. Assuming that f and g are differentiable functions, find the value of if f (4) = 5, f '(4) = –7, g(4) = 8, and g'(4) = 4.

337.

338.

339.

340.

341.

342.

343. Assuming that f and g are differentiable functions, find the value of if f (5) = –4, f '(5) = 2, g(5) = –7, and g'(5) = –6.

344.

345.

346.

347.

348.

349.

350. Assuming that g is a differentiable function, find an expression for the derivative of .

351. Assuming that g is a differentiable function, find an expression for the derivative of .

Using the Chain Rule to Find Derivatives

352–370 Use the chain rule to find the derivative.

352.

353.f (x) = sin(4x)

354.

355.

356.

357.

358.

359.f (x) = cos(x sin x)

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360.

361.

362.f (x) = sec² x + tan² x

363.

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364.

365.

366.

367. , where x > 1

368.

369.

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370.

More Challenging Chain Rule Problems

371–376 Solve the problem related to the chain rule.

371. Find all x values in the interval [0, 2π] where the function f (x) = 2 cos x + sin² x has a horizontal tangent line.

372. Suppose that H is a function such that for x > 0. Find an expression for the derivative of .

373. Let , g(2) = –2, g'(2) = 4, f'(2) = 5, and f'(–2) = 7. Find the value of F'(2).

374. Let , f (2) = –2, f'(2) = –5, and f'(–2) = 8. Find the value of F'(2).

375. Suppose that H is a function such that for x > 0. Find an expression for the derivative of f (x) = H(x³).

376. Let , g(4) = 6, g'(4) = 8, f '(4) = 2, and f '(6) = 10. Find the value of F '(4).